IMPEDANCE CONCEPT AND APPLICATION 23 



pedance" will be used to designate a constant of the medium without 

 reference to any particular wave. 



Vibrating Strings 



In strings under constant tension r, simply periodic waves may be 

 described by the following two equations : 



dF / , • N dv too „ 



-7— = — [r -\- tcotnw, ^- = p, 



ax dx T 



where m is the mass and r the resistance per unit length of the string. 

 The variable F represents the force on a typical point of the string at 

 right angles to the string and v is the velocity at that point. 



Hence the characteristic impedance and the propagation constant 

 are given by 



[(F+lomtJT r ~r Tico 



Zo = x- -. , r = -y/(r + icom) — . 



\ 4co \ r 



In the non-dissipative case we have simply 



y ! ' T • f^ 



Heat Waves 



Transmission of heat waves is also a special case of the generalized 

 transmission line theory. In the one-dimensional case we have 



dT _ V dv _ dT 



"dx^ ~K' dx~ ~ ^ 'm' 



where: T is the temperature, v the rate of heat flow, K the thermal 

 conductivity, 5 the density and c the specific heat. For simply peri- 

 odic waves, we obtain 



dT __1 ^ — — • ?i T 



dx K ' dx 



Thus the characteristic impedance and the propagation constant of 

 heat waves are 



1 „ jioocd 



Zn — 



^icocdK ' \ K ' 



The ratio "the temperature of the source/the rate of heat flow from 

 the source" is the impedance "seen" by the heat source. 



